Hey, Melissa! Over the years, our general rule for our four kids is that if the child is school-aged then s/he is expected to walk through the parks. We certainly take frequent breaks the younger they are -- taking time out for cool drinks and park benches or ducking into attractions with seating and air conditioning.
If your boys are in good health, I would take a "wait-and-see" approach with the strollers. There
are sturdy double-strollers available for rental at each parks' Guest Services (located at or near the entrance). Yet, the daily rental of strollers -- not to mention the wear and tear on whomever does the pushing -- may not be worth the trouble in the end. Plus, as crowded as the parks are in June, you may find that being able to move quickly on foot rather than with a stroller is more desirable.
Of course, you know your children best, but my advice would be to talk about the walking with the boys beforehand and perhaps try out their endurance level at a local park or zoo. We've been taking or renting one or more strollers at WDW for more than a decade, and we were actually relieved when our six-year-old said that he didn't need one
last year! Have a wonderful day and a magical vacation!